Demazure crystals for flagged key polynomials

TL;DR

This paper extends crystal structures to flagged key tableaux, demonstrating their Demazure crystal structure and connecting them to flagged key polynomials through a generalized insertion algorithm.

Contribution

It introduces a new crystal structure on flagged key tableaux and establishes their Demazure crystal properties, linking combinatorial objects to polynomial representations.

Findings

Flagged key tableaux possess a natural Demazure crystal structure.

The generalized weak Edelman-Greene insertion provides a bijection with crystal isomorphism.

Flagged key polynomials are recovered as characters of these Demazure crystals.

Abstract

One definition of key polynomials is as the weight generating functions of key tableaux. Assaf and Schilling introduced a crystal structure on key tableaux and related it to Morse--Schilling crystals on reduced factorizations for permutations via weak Edelman--Greene insertion. In this paper, we consider generalizations of both crystals depending on a flag. We extend weak EG insertion to a bijection between our flagged objects and show that the recording tableau gives a crystal isomorphism. As an application, we show that flagged key tableaux have a natural Demazure crystal structure, whose characters recover Reiner and Shimozono's flagged key polynomials.